Suppose
X
∼
N (0
,
1)
, and let
Y
=
X
2
. Find the cdf and pdf of
Y
in terms of
φ
(the cdf of
X
).
Let Z ∼ N (0, 1), and let X = max(Z, 0). 1. Find FX in terms of Φ(t). Is X a continuous random variable ? 2. Compute p(X = 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that limt t nφ(−t) = 0 for all n ≥ 0. 4. Find the CDF FX2 (u) 5. Compute V(X). Hint: use FX2 , and follow the same hint of part (3)
Let X be a random variable with pdf S 4x3 0 < x <1 Let Y 0 otherwise f(x) = {41 = = (x + 1)2 (a) Find the CDF of X (b) Find the pdf of Y.
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0, 1). Show that the distribution of Q = X/Y follows the Cauchy distribution, i.e., f(q) = 1/π(1+q2) . Hint: Let Q = X/Y and V=Y. Find the joint pdf of Q and V and finally find the marginal pdf of Q by integrating the joint pdf of Q and V w.r.t. V: Y π(1+q2) Y V = Y . Find the joint pdf of...
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Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute V(X) (Hint: use fxa found above
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute...
7. Let X and Y have joint pdf 122 (1-x)y, 0, 0〈x〈1,0くyく1. otherwise. x,y(x,y) = (a) Find the joint cdf of X and Y. (4pts) (b) Find PY< VX. (Spts) (c) Find the marginal pdfs of X and Y. (6pts) (d) Are X and Y independent? (5pts)
7. Let X and Y have joint pdf 122 (1-x)y, 0, 0〈x〈1,0くyく1. otherwise. x,y(x,y) = (a) Find the joint cdf of X and Y. (4pts) (b) Find PY
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)
1) Let X and Y have joint pdf: fxy(x,y) = kx(1 – x)y for 0 < x < 1,0 < y< 1 a) Find k. b) Find the joint cdf of X and Y. c) Find the marginal pdf of X and Y. d) Find P(Y < VX) and P(X<Y). e) Find the correlation E(XY) and the covariance COV(X,Y) of X and Y. f) Determine whether X and Y are independent, orthogonal or uncorrelated.
Let X be a continuous random variable, and let Y = g(X), where g(x) = { 0; x < 0 or x > 3 { x; 0 <= x < 1 { 1; 1 <= x < 2 { 3-x; 2 <= x <= 3 (a) Express the cdf of Y in terms of the cdf of X. (b) Determine and sketch the cdf of Y when X is exponentially distributed with parameter α = 1. Is the cdf of...
Let X and Y ~U(0, 1]. X and Y are independent a) Find the PDF of X+Y b) Suppose now X~(0, a] Y~(0,b] and . Find the PDF of X+Y Ο <α<b
0 Sy s 1. Let X and Y have joint pdf: fx,y(x, y) = kx(1 – x)y for 0 < x < 1, (a) Find k. (b) Find the joint cdf of (X,Y). (c) Find the marginal pdf of X and of Y. (d) Find Pſy < 81/2],P[X<Y]. (e) Are X and Y independent? (f) Find the correlation and covariance of X and Y. (g) Determine whether X and Y are uncorrelated. (h) Find fy(y|x) (i) Find E[Y|X = x]...